I wish to say "I understand you will be looking after my work permit application."
I wondered if I should use "kümmert sich nach" rather than "kümmert sich um"
Ich verstehe Sie werden sich nach / um meine Arbeitsbewilligung kümmeren.

Kümmern um is correct then (in a mail to an official, "kümmern" would be much to casual). Do you need the "I understand..." or is that you an opening bla bla? If you it is important that we need to rephrase the sentence a bit because like it is now, the sentence means "I do understand that you will take care of it, I am not that stupid". I'm sure that's not what you try to convey.